On b-edge consecutive edge labeling of some regular trees

Let G = (V,E) be a finite (non-empty), simple, connected and undirected graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection α from V ∪ E to the integers 1, 2, . . . , n+ e, with the property that for every xy ∈ E, α(x) + α(y) + α(xy) = k, for some constant k. Such a labeling is called a b-edge consecutive edge magic total if α(E) = {b + 1, b + 2, . . . , b + e}. In this paper, we proved that several classes of regular trees, such as regular caterpillars, regular firecrackers, regular caterpillar-like trees, regular path-like trees, and regular banana trees, have a b-edge consecutive edge magic labeling for some 0 < b < |V |.


Introduction
All graphs which are considered in this paper are finite, simple, connected and undirected. Let G = (V, E) be a graph with vertex set V and edge set E. The labeling α : V ∪ E → {1, 2, ..., |V | + |E|} of G is called edge magic total if every edge xy has the same weight w(x) = α(x)+α(y)+α(xy) = k, and G is called an edge magic total graph if an edge magic total labeling of G exists. If α(V ) = {1, ..., n} then α is called a super edge magic total labeling. Magic labeling introduced by Sedláček [8] in 1963, and until now, the research is grown and there are many results in magic labeling, especially in edge magic labeling. There are some results on some classes of trees, such as banana trees [5]. The super edge magic strength of caterpillars, firecrackers and banana trees were studied by Swaminathan and Jeyanthi [11]. For further results in graph labeling, including the (super) edge magic total labeling, wecan see [4].
A bijection f : V (G) ∪ E(G) → {1, 2, ..., |V | + |E|} is called a b-edge consecutive edge magic total labeling of G = G(V, E) if f is an edge magic total labeling and f (E) = {b + 1, ..., b + e}, 0 ≤ b ≤ n. A graph G that has b-edge consecutive edge magic total labeling is called a bedge consecutive edge magic total graph. For simplicity, for the rest of the paper, we will use b-ECEMTL for an abbreviation of b-edge consecutive edge magic total labeling. Since if b = 0 the 0-ECEMTL will be a well known super edge magic total labeling, which are already studied by many researchers, and in the case b = n the labeling can be found by dual of super edge magic total labeling (if any), then in this paper, we only consider the case of 0 < b < n. The most famous conjecture on edge magic labeling area is from Enomoto et al. [3], which is "every tree is super edge magic graph." This conjecture might also be true for b-ECEMTL. On the direction of showing the conjecture is true, in this paper, we study several classes of regular trees, such as regular caterpillars, regular firecrackers, regular caterpillar-like trees, regular path-like trees and regular banana trees.

Known Results
Sugeng and Miller introduced the concept of b-ECEMTL in 2008. This paper was inspired by the concept of the edge consecutive vertex magic total labeling and vertex consecutive vertex magic total labeling by Balbuena et al. [1]. Sugeng and Miller [9] proved several results as follows. The first theorem said that we always can find a graph that has b-ECEMTL for every b, 0 < b < n.

Regular Caterpillar and Regular Firecracker
A caterpillar is a graph derived from a path by hanging any number of leaves from the vertices of the path. If the number of leaves of every center the same, then we called it a regular caterpillar. We call the path which its vertices are the centers of the caterpillar as a backbone path of the caterpillar. A firecracker is a graph obtained from the concatenation of stars by linking one leaf from each. We call the linking leaf as a backbone path of the firecracker. If a firecracker is obtained from the concatenation of isomorphic stars, we get a regular firecracker. A caterpillar can be obtained from firecracker by moving the edges linking one leaf from each star S i to linking each center of S i and vice versa. Theorem 2.3 gives the result that every caterpillar has a b-ECEMTL for some b ∈ (0, |V |). The similar result has done by Kang et al. [6] that caterpillar has a b-ECMTL for some specific b. They also proved that if G is a tree with the bipartite set V (G) = V 1 ∪ V 2 and having a b-edge consecutive magic labeling then b ∈ {0, |V 1 |, |V 2 |, |V |}. However, in their paper they also included the value b = 0 and b = |V |, which we do not consider in this paper. The preliminary results in this subsection already presented in [10]. It is known that every caterpillar has a b-ECEMTL (Theorem 2.3 and in [6]). However, in the following theorem, we give an alternative proof for the regular caterpillar.
Proof. Let G be a regular caterpillar with c i as its center vertices, for i = 1, 2, ..., k and r as the number of leaves of every center. Let v j i be the j-th leaf of the center c i , i = 1, ..., k and j = 1, ..., r. Since caterpillar is a bipartite graph, then we can divide the set of its vertices as two disjoint sets of vertices, say V 1 and V 2 . Arrange the vertices such that if we draw the caterpillar, then the edges do not intersect each other. As an example we can put the center c 1 in the set V 1 and label it with 1, and put the leaf vertices of the center v j 1 , j = 1, ..., r in the set V 2 and label it with b + 1, ..., b + k. The next step, put the leaves of center c 2 , v j 2 , j = 1, ..., r, in the set V 1 and label it with 2, ..., 2 + r − 1, then put the center vertex c 2 in the set V 2 and label it with b + r + 1. This process can continue until all vertices have its label.
The weight f (u) + f (v) for every edge uv in the caterpillar will form consecutive integers. By completing the edge label with b + 1, b + 2, ..., b + |E|, following the edge weight starts from the edge with the biggest label, then we can see that f is a b-ECEMTL, with b = |V 1 | .
, , for all i = 1, ..., k, to return the graph to the firecracker form.
The moving process of the edges c i c i+1 to v * i v * i+1 , for all i = 1, ..., k, in step (iii) guarantee the b-ECEMTL for the firecracker.

Path-like and Caterpillar-like trees
Let P n be a path with n vertices. Embed the path in the two dimensional grid where the vertex is located in the intersection point of the grid. An elementary transformation of the path is a process by replacing the edge xy by a new edge x * y * , such that the edge weight set does not change. A tree T of order n is called path-like tree when it can be obtained from embedding a path in the two-dimensional grid and using set of elementary transformations. The structure of the path-like tree was studied by Muntaner-Batle and Rius-Font [7]. Later, Sugeng and Silaban in [10] use generalisation of the path-like tree on a backbone path of caterpillar to obtain a super edge magic total labeling on new subclass of trees that they called caterpillar-like trees. This idea can be use for the regular caterpillar to obtain a b-ECEMTL for regular caterpillar-like trees. The super edge magic strength of caterpillar and firecracker was studied by Swaminathan and Jeyanthi [11]. Figure 1 gives the example of b-ECEMTL for the regular caterpillar-like tree. Theorem 2.6. Every regular caterpillar-like tree has a b-ECEMTL, where 0 < b < n.
Proof. Label the regular caterpillar-like tree using the label in Algorithm 2.
1. Label the regular caterpillar with b-ECEMTL given in the proof of Theorem 3.
2. Remove all the labeled leaves from the vertices of the backbone path. 3. Embed the backbone path of the labeled caterpillar in the two-dimensional grid. 4. Do some elementary transformation on the backbone path by replacing the edge by a new edge 5. Put back all labeled leaves to the associated vertices of the path-like tree.
The elementary transformation in step (iv) keeps the b-ECEMTL property of the new graph.
Corollary 2.1. All regular path-like trees have a b-ECEMTL, where 0 < b < n.

Regular Banana Trees
A regular (k, r)-banana tree is a graph obtained by connecting one leaf of each of k copies of a star S r graph with a single root vertex that is distinct from all the stars [2]. Theorem 2.7. For r ≥ 3, every regular banana tree B(r, r) has an r 2 -ECEMTL.
Proof. Let a be the root vertex of the regular banana tree B(r, r). Let c 1 , ..., c r be the center of the star and v j i be the j-th vertex of i-th star, i, j = 1, ..., r. Let v 1 i be a vertex which is adjacent to the root, for i = 1, ..., r.
Label the regular banana tree using the label in Algorithm 3.